Bounding the optimal rate of the ICSI and ICCSI problem

In this work we study both the index coding with side information (ICSI) problem [5] and the more general problem of index coding with coded side information (ICCSI), as introduced by Shum et al in 2012. We investigate the optimal rate of an instance of the index coding problem. In the ICSI problem case, we characterize those digraphs having min-rank one less than their order and we give an upper bound on the minrank of a hypergraph whose incidence matrix can be associated with that of a 2-design. Some security aspect are discussed in the particular case when the design is a projective plane. For the coded side information case, we extend the graph theoretic upper bounds given by Shanmugam et al in 2014 on the optimal rate of index code.